Multistability of partially synchronous regimes in a system of three coupled logistic maps

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摘 要:

We research the bifurcations of partially syn- chronous periodic orbits which lead to the multistability formation in a system of three coupled logistic maps with symmetrical diffusive coupling. We demonstrate that the par- tially synchronous regimes appeare as results of saddle-node bifurcations. The mechanisms of stability loss of partial syn- chronization of chaos are discussed. I. INTRODUCTION Nonlinear oscillatory systems often demonstrate the phe- nomenon of multistability, when several attractors coexist in the phase space at the same parameters values. Such systems attract a great interest of researchers because of their promising possibilities for the aim of control of oscillatory dynamics. The typical example of a system with developed multistability is a small ensemble of diffusively coupled period-doubling oscillators. As it was shown in a number of works (1)-(3), two coupled period-doubling oscillators demonstrate a great variety of multistable regular and chaotic regimes. In our previous investigations, we have found that the mechanism of the multistability formation in such sys- tems is connected with the phenomenon of loss of complete chaotic synchronization (4). The process of the synchronism breaking at parameters change is appeared to be induced by the same bifurcations of the principal periodic solutions that lead to the formation of new stable regimes in the systems phase space. Is this situation typical for the systems with higher dimension? How the process of multistability formation interacts with the synchronization phenomenon in ensembles of more than two oscillators? In attempt to answer these questions we consider an ensemble with one more oscillator: a ring of three coupled period-doubling maps. It demonstrates phenomena of both complete and partial chaotic synchronization. The last case denotes that only two oscillators in the ring are synchronized while the behavior of the third one remains unsynchronous. Interdependence between complete and partial synchronization in systems of three oscillators with differnet types of coupling, bifurcations that lead to the synchronism breaking, as well as the phenom- ena of bubbling of the attractors and the riddled basins which accompany them, have been considered in a number of works (5)-(9). In our investigations we concentrate on the role of this bifurcations in the process of multistability formation. We describe typical chains of bifurcations which lead to formation of hierarchy of partially synchronous regimes.